Optimal. Leaf size=160 \[ \frac{494}{567} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )-\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{63 \sqrt{3 x+2}}+\frac{2470}{567} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{2209}{567} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0504041, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{63 \sqrt{3 x+2}}+\frac{2470}{567} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{494}{567} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2209}{567} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{2}{9} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\\ &=-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{4}{189} \int \frac{\left (\frac{1395}{4}-\frac{3705 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{2470}{567} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}-\frac{4 \int \frac{-\frac{5865}{2}-\frac{33135 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1701}\\ &=\frac{2470}{567} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{2209}{567} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx-\frac{2717}{567} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2470}{567} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{63 \sqrt{2+3 x}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}-\frac{2209}{567} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{494}{567} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.25706, size = 102, normalized size = 0.64 \[ \frac{-10360 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (1575 x^2+2841 x+1187\right )}{(3 x+2)^{3/2}}+2209 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1701} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 224, normalized size = 1.4 \begin{align*}{\frac{1}{17010\,{x}^{2}+1701\,x-5103} \left ( 31080\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6627\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+20720\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -4418\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +94500\,{x}^{4}+179910\,{x}^{3}+59916\,{x}^{2}-44016\,x-21366 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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